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← JEE Main 2021 · 24 Feb · Shift 1
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JEE Main 2021 · 24 Feb · Shift 1, Mathematics question 87

If ∫−aa(∣x∣+∣x−2∣)dx=22\displaystyle\int_{-a}^a(|x|+|x-2|)dx=22, (a > 2) and [x] denotes the greatest integer ≤ x, then ∫a−a(x+[x])dx\displaystyle\int_a^{-a}(x+[x])dx is equal to _______.

Answer and worked solution

Official answer: 3

Find a. ∫−aa∣x∣ dx=a2\int_{-a}^a|x|\,dx=a^2 and ∫−aa∣x−2∣ dx=a2+4\int_{-a}^a|x-2|\,dx=a^2+4, so 2a2+4=222a^2+4=22, a=3a=3.

Integral. ∫3−3(x+[x])dx=−∫−33[x]dx=−(−3−2−1+0+1+2)=3\int_3^{-3}(x+[x])dx=-\int_{-3}^3[x]dx=-(-3-2-1+0+1+2)=3.

Answer: 3

Topics:Properties of definite integrals
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